Signal processing method, signal processing program, recording medium storing the signal processing program and signal processor

ABSTRACT

A signal processing method is disclosed, where, after inputting a measurement data weighted-spline filter formula is selected to calculate an initial value of a spline filter, weight is adjusted, spline filter output is calculated, convergence is judged thereafter. When the weight is not judged converged, the weight is updated and the weight adjusting and spline filter output calculation are repeated, to conduct a robust spline filtering on the measurement data.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a signal processing method, morespecifically, relates to a signal processing method for filtering a dataobtained by measuring surface texture of an object to be measured suchas roughness or a contour etc. of the object to be measured.

2. Description of Related Art

Measuring instruments for measuring surface texture of an object to bemeasured e.g. contour, roughness, waviness, etc. of a surface of theobject to be measured such as a coordinate measuring machine formeasuring a three-dimensional shape of the object to be measured, a formmeasuring machine or an image measuring machine for measuring atwo-dimensional contour, a roundness measuring machine for measuringroundness, and further, a surface roughness tester for measuringwaviness and roughness etc. of the surface of the object to be measuredhave been conventionally known. In such measuring instruments a contacttype or non-contact type sensor and the object to be measured are movedrelatively, to collect measurement data of the surface of the object tobe measured.

Usually, disturbances such as noise are contained in thus collectedmeasurement data.

Since the disturbance is usually an electric and magnetic inductionnoise etc. containing high-frequency components, when the contour of thesurface of the object is to be measured, the surface roughness and thewaviness can be the disturbance of the contour.

In order to remove the disturbance as necessary, the measurement data isoften filtered to remove, for example, the high-frequency components. Aspline filter is known as such a filter (refer to Document 1: JapanesePatent Laid-Open Publication No. HEI 9-179992, and Document 2: JapanesePatent Laid-Open Publication No. HEI 8-278343). Further, a robust splinefilter is known for applying weight on each measurement data and forrepeating to update of the weight until this weighting is optimized.

However, the robust spline filter has been applicable to curve data, butit has not been applicable to curved surface data obtained by measuringa surface area.

SUMMARY OF THE INVENTION

A primary object of the present invention is to provide a signalprocessing method having a high reliability, which allows data obtainedby measuring the surface area to be utilized, maintaining a distinctivefeature of the spline filter, and is also capable of being applied tosurface measurement data such as the surface texture of the object to bemeasured, and in addition hereto, to provide a signal processingprogram, a recording medium storing the signal processing program and asignal processor.

A signal-processing method according to an aspect of the presentinvention is for filtering a measurement data obtained by measuring thesurface area the method including: a measurement data input step forinputting the measurement data; a selecting step for selecting apredetermined weighted spline filter formula; an initializing step forapplying weight on the measurement data by a unit matrix to obtain aninitial value of a spline filter output; a weight-adjusting step foradjusting and determining the weight on the measurement data; a splinefilter output calculating step for calculating the spline filter outputusing the weight determined in the weight-adjusting step; a convergencetest step for determining a convergence of the weight; and an outputstep for outputting a signal-processing result based on the splinefilter output, wherein, when it is not judged that the weight isconverged in the convergence judging step, the weight is updated and theweight-adjusting step and the spline-filter output-calculating step arerepeated and wherein a smoothing spline curved surface corresponding tothe measurement data is generated, thereby to conduct a robust splinefiltering on the measurement data.

In accordance with this invention, the weighted-spline filter formula isselected and the spline curved surface as a spline filtering output isrepeatedly calculated while sequentially updating its weight based uponthe selected spline filter formula, so that the robust-spline filteringhaving the filter output as the signal processing result of the splinecurve when the weight is converged can be applied on the measurementdata. Accordingly a deformation at the start or end area of themeasurement data (an end-effect) can be prevented, and the shapecontained in the measurement data can be extracted without beinginfluenced by trackability for waviness component and the noisecomponent in the measurement data. As a result, the filtering withexcellent shape trackability can be conducted thereby further enhancingthe reliability of the measurement data.

Herein, the measurement data obtained by measuring the surface arearefers to, for example, data etc. obtained by scanning the surface ofthe object to be measured with the coordinate measuring machine or thesurface texture measuring machine. That is, it refers to data to berepresented with height data f (x, y) from the surface with respect to asampling position (x, y).

In the signal-processing method according to the above aspect of thepresent invention, the weight determined in the weight-adjusting step ispreferably adjusted to smaller value as a deviation of the measurementdata from the spline curve calculated by the weighted-spline filterformula becomes greater.

According to the above arrangement, since the weight of the measurementdata is adjusted smaller as the deviation of the measurement data fromthe spline curve calculated by the weighted-spline filter formulabecomes greater, robust-spline filtering without being influenced bysingular point data contained in the measurement data is provided.Specifically the spline curve is repeatedly calculated while applyingsmall weight on the measurement data remote from the spline curve andapplying heavy weight on the measurement data near the spline curve.Then, the spline curve is gradually approximated to the original shapecomponent (for example, a true shape value etc. of the object to bemeasured) contained in the measurement data. And, the final spline curvewhen the weight is judged converged is regarded as a shape componenthaving sufficiently small error relative to the original shapecomponent. As a result, the extremely accurate robust-spline filteringcan be conducted.

In the signal-processing method according to the above aspect of thepresent invention, the weight is preferably judged converged when amountof weight-change in the weight adjusting step becomes a predeterminedvalue or less in the convergence judging step,

According to the above arrangement, since the weight is judged convergedwhen the change in the weight during the repeated-loop process becomes apredetermined value or less, increase in a processing time on account ofthe unnecessary repetitive loop can be avoided and time required forrobust spline filter processing can be reduced. Further, since the errorof spline curve is considered small enough relative to the originalshape component contained in the measured data when amount ofweight-change in the weight adjusting step becomes a predetermined valueor less, extremely accurate robust-spline filtering can be conducted.

In the signal-processing method according to the above aspect of thepresent invention, the output step preferably includes: aweight-updating step for updating the weight to 1 when the weight of themeasurement data exceeds a predetermined value, a spline filterre-output calculating step for calculating the spline filter output,based upon the updated weight, and a signal process result output stepfor outputting the spline filter output in the spline filter re-outputcalculating step as a signal processing result.

According to the above arrangement when the weight of the measurementdata exceeds a predetermined value the weight is updated to 1 and whenthe weight of the measured data is not greater than a predeterminedvalue the weight is updated to 0, and the spline filter output isre-calculated to be outputted as a signal processing result.

Specifically when the weight is judged converged after repeating theweight adjusting step and the spline filter output calculating step, themeasurement data where its weight exceeds a predetermined value isconsidered as an efficient data with its weight being updated to 1,however, the measurement data where its weight is not greater than apredetermined value is considered as an inefficient data with its weightbeing updated to 0, and the spline filter output after the updating isobtained again so that the robust spline filter processing can be moresecurely conducted. Then, since the result is outputted as thesignal-processing result, a spline curve with sufficiently small errorrelative to the original shape component contained in the measurementdata can be calculated. Consequently, robust spline filtering withexcellent shape trackability can be achieved.

Also, in the present invention, the measurement data inputting steppreferably includes a step of deleting singular point data separatedlocally as against the measurement data.

In accordance with this invention, due to, for example, the stronginduced noise occurred from power appliances located in a factory etc.,data which has been locally protuberant and has been separated, forexample, data which is extremely different from that of data on bothsides can be pre-deleted as obviously singular point data, so areliability of the robust-spline filter process is further enhanced.

The signal process program of the present invention causes a computer toexecute the above-mentioned signal processing method. Also, therecording medium of the present invention has the above-mentioned signalprocess program recorded computer-readably. And, the signal processingunit of the present invention causes a computer to execute theabove-mentioned signal processing program.

With the configuration as described above, given that the computerhaving a CPU (Central Processing Unit) and a memory (memory device) isbuilt-in the signal processing unit and the program is configured so asfor computer to execute each step, parameters can be easily change, forexample, various parameters including decision of the separationquantity according to the dimension of the measurement data in additionto the weight adjustment or the convergence determination etc. And, therecording medium with this program recorded therein may be directlyinserted into the computer, thereby to install the program in thecomputer, and a read device for reading out information of the recordingmedium may be externally attached to the computer, thereby to installthe program in the computer from this read device. Additionally, theprogram may be supplied to and installed in the computer viacommunication lines such as an Internet, a LAN cable, and a telephonecircuit, or a wirelessly.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart showing signal-processing process in asignal-processing method according to a first reference technology;

FIG. 2 is a functional block diagram of a signal-processing device inthe first reference technology;

FIG. 3A and FIG. 3B are illustrations comparing a result of splineprocessing and a result of robust spline processing for one-dimensionaltime-series data in the first reference technology;

FIG. 4A and FIG. 4B are illustrations comparing a result of splineprocessing and a result of robust spline processing in a third referencetechnology;

FIG. 5 is an illustration showing transfer characteristic of thesignal-processing method according to the present invention;

FIG. 6 is a flowchart showing a modification of the present invention.

FIG. 7 is a view illustrating an example of measurement data to beapplied for a first embodiment;

FIG. 8 is a view illustrating a result obtained by applying the robustspline filter on the measurement data to be shown in FIG. 7 in the firstembodiment;

FIG. 9 is a view illustrating a result obtained by processing themeasurement data to be shown in FIG. 7 with a Gaussian filter;

FIG. 10 is a view illustrating a profile in the X direction in FIG. 7 inthe first embodiment;

FIG. 11 is a view illustrating a profile in the Y direction in FIG. 7 inthe first embodiment;

FIG. 12 is a view illustrating one example of a transfer characteristicin the first embodiment; and

FIG. 13 is a flowchart illustrating a signal process in the firstembodiment relating to the signal processing method.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT(S)

The signal processing method of the present invention primarily relatesto a signal processing method of performing a filter process for thecurved surface data obtained by measuring the surface area; however therobust spline filter (spline filter) for the curve data is shown as areference technology on the premise of explaining the present invention.

<First Reference Technology>

Initially, weighted spline filter will be described below.

As an example, when the number of data is represented by n, measurementdata is represented by y_(k) (k=0, 1, . . . , n−1) and spline functionis represented by s, a spline filter is obtained by minimizing sum ofsquares of residual error relative to measurement data

$\begin{matrix}{\sum\limits_{k = 0}^{n - 1}\left( {y_{k} - {s\left( x_{k} \right)}} \right)^{2}} & (1)\end{matrix}$under the condition that spline energy

$\begin{matrix}{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}{\mathbb{d}x}}} & (2)\end{matrix}$is minimized.

In other words, when I(s) is represented by the following formula,

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}} + {\lambda{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}{\mathbb{d}x}}}}}} & (3)\end{matrix}$the spline filter is obtained by minimizing the I(s) (where λ representsLagrange constant).

When weight for the residual error at respective measurement point isrepresented by w_(k) (k=0, 1, . . . , n−1), following formula adapted toweighted spline filter can be obtained.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}{w_{k}\left\{ {y_{k} - {s\left( x_{k\;} \right)}} \right\}^{2}}} + {\lambda{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}{\mathbb{d}x}}}}}} & (4)\end{matrix}$

The spline function s is digitized at a constant pitch where the secondterm is modified as

$\begin{matrix}{\alpha{\sum\limits_{k = 0}^{n - 1}{\nabla^{2}{s\left( x_{k} \right)}}}} & (5)\end{matrix}$

Then, following equation is established.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}{w_{k}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}}} + {\alpha{\sum\limits_{k = 0}^{n - 1}{\nabla^{2}{s\left( x_{k} \right)}}}}}} & (6)\end{matrix}$

In the above:∇² s(x _(k))=s(x _(k+1))−2s(x _(k))+s(x _(k−1))  (7)

Accordingly, a value s_(k) of discrete spline for minimizing I(s)satisfies the following formula.

$\begin{matrix}{{\frac{\partial{I\left( {s_{0},s_{1},{\cdots\mspace{14mu} s_{n - 1}}} \right)}}{\partial s_{k}} = 0}{{k - 0},1,\cdots\mspace{14mu},{n - 1}}} & (8)\end{matrix}$

The weighted spline filter is defined as a spline function forminimizing I(s) in the formula (6).

In order to represent the weighted spline filter for non-periodicmeasurement data as a matrix, when a boundary condition of non-periodicmeasurement is:∇² s(x ₀)=0 ∇² s(x _(n−1))=0  (9)

since following formula,

$\begin{matrix}{{\frac{\partial I}{\partial s_{0}} = {{{- 2}{w_{0}\left( {y_{0} - s_{0}} \right)}} = {2{\alpha\left( {s_{2} - {2s_{1}} + s_{0}} \right)}}}}{\frac{\partial I}{\partial s_{1}} = {{{- 2}{w_{1}\left( {y_{1} - s_{1}} \right)}} + {2{\alpha\left( {s_{3} - {4s_{2}} + {5s_{1}} - {2s_{0}}} \right)}}}}{\frac{\partial I}{\partial s_{k}} = {{{- 2}{w_{k}\left( {y_{k} - s_{k}} \right)}} + \mspace{76mu}{2{\alpha\left( {s_{k + 2} - {4s_{k + 1}} + {6s_{k}} - {4s_{k - 1}} + s_{k - 2}} \right)}}}}{{k = 2},3,\cdots\mspace{14mu},{n - 3}}{\frac{\partial I}{\partial s_{n - 2}} = {{{- 2}{w_{n - 2}\left( {y_{n - 2} - s_{n - 2}} \right)}} + \mspace{101mu}{2{\alpha\left( {s_{n - 4} - {4s_{n - 3}} + {5s_{n - 2}} - {2s_{n - 1}}} \right)}}}}{\frac{\partial I}{\partial s_{n - 1}} = {{{- 2}{w_{n - 1}\left( {y_{n - 1} - s_{n - 1}} \right)}} + {2{\alpha\left( {s_{n - 3} - {2s_{n - 2}} + s_{n - 1}} \right)}}}}} & (10)\end{matrix}$is established by substituting Q as

$\begin{matrix}{Q = \begin{pmatrix}1 & {- 2} & 1 & \; & \; & \; & \; \\{- 2} & 5 & {- 4} & 1 & \; & \; & \; \\1 & {- 4} & 6 & {- 4} & 1 & \; & \; \\\; & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \; \\\; & \; & 1 & {- 4} & 6 & {- 4} & 1 \\\; & \; & \; & 1 & {- 4} & 5 & {- 2} \\\; & \; & \; & \; & 1 & {- 2} & 1\end{pmatrix}} & (11)\end{matrix}$the matrix of the weighted spline filter for non-periodic data can beexpressed as

$\begin{matrix}{{\left( {W + {\alpha\; Q}} \right)S} = {{WY}\mspace{14mu}{where}}} & (12) \\{W = \begin{pmatrix}w_{0} & \; & \; & \; & \; & \; \\\; & w_{1} & \; & \; & \; & \; \\\; & \; & ⋰ & \; & \; & \; \\\; & \; & \; & w_{n - 3} & \; & \; \\\; & \; & \; & \; & w_{n - 2} & \; \\\; & \; & \; & \; & \; & w_{n - 1}\end{pmatrix}} & (13) \\{S = {{\begin{pmatrix}s_{0} \\s_{1} \\\vdots \\s_{n - 3} \\s_{n - 2} \\s_{n - 1}\end{pmatrix}\mspace{14mu} Y} = \begin{pmatrix}y_{0} \\y_{1} \\\vdots \\y_{n - 3} \\y_{n - 2} \\y_{n - 1}\end{pmatrix}}} & \;\end{matrix}$

Next, in order to represent the weighted spline filter for periodicmeasurement data as a matrix, when periodic boundary condition of theperiodic measurement data is represented as

$\begin{matrix}\begin{matrix}{s_{k + n} = s_{k}} & \; & {{k = 0},1,\cdots\mspace{11mu},{n - 1}}\end{matrix} & (14) \\{since} & \; \\{\frac{\partial I}{\partial s_{k}} = {{{- 2}\;{w_{k}\left( {y_{k} - s_{k}} \right)}} + {2\;{\alpha\left( {s_{k + 2} - {4\; s_{k + 1}} + {6\; s_{k}} - {4\; s_{k - 1}} + s_{k - 2}} \right)}}}} & (15) \\{{k = 0},1,\cdots\mspace{11mu},{n - 1}} & \; \\{for} & \; \\{\overset{\sim}{Q} = \begin{bmatrix}6 & {- 4} & 1 & \; & \; & 1 & {- 4} \\{- 4} & 6 & {- 4} & 1 & \; & \; & 1 \\1 & {- 4} & 6 & {- 4} & 1 & \; & \; \\\; & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \; \\\; & \; & 1 & {- 4} & 6 & {- 4} & 1 \\1 & \; & \; & 1 & {- 4} & 6 & {- 4} \\{- 4} & 1 & \; & \; & 1 & {- 4} & 6\end{bmatrix}} & (16)\end{matrix}$the matrix of the weighted spline filter for periodic measurement datacan be represented as(W+α{tilde over (Q)})S=WY  (17)

The amplitude characteristic and phase characteristic of the splinefilter at steady state will be described below.

Following spline filter formulay _(k) =S _(k)+α(s _(k+2)−4s _(k+1)+6s _(k)−4s _(k−1) +s _(k−2))k=0,1, .. . ,n−1  (18)where weight W=I (unit matrix), is expressed by z-transform asy _(k) =s _(k)+α(z ⁻²−4z ⁻¹+6−4z+z ²)s _(k)  (19)

Transfer function H(z) of the spline filter is represented by

$\begin{matrix}\begin{matrix}{{H(z)} = \frac{s_{k}}{y_{k}}} \\{= \frac{1}{1 + {\alpha\left( {z^{- 2} - {4\; z^{- 1}} + 6 - {4z} + z^{2}} \right)}}}\end{matrix} & (20)\end{matrix}$

In order to examine the amplitude characteristic and phasecharacteristic, Z is substituted asz=e^(jωΔx)   (21)so that following formula is established.

$\begin{matrix}{{H(\omega)} = \frac{1}{1 + {\alpha\left( {{\mathbb{e}}^{{- 2}\; j\;\omega\;\Delta\; x} - {4\;{\mathbb{e}}^{{- j}\;\omega\;\Delta\; x}} + 6 - {4\;{\mathbb{e}}^{j\;\omega\;\Delta\; x}} + {\mathbb{e}}^{j\;\omega\;\Delta\; x}} \right)}}} & (22) \\{Since} & \; \\{{{\mathbb{e}}^{{- j}\;\alpha} + {\mathbb{e}}^{j\;\alpha}} = {2\mspace{11mu}\cos\mspace{11mu}\alpha}} & (23)\end{matrix}$following equation is established.

$\begin{matrix}{{{\mathbb{e}}^{{- 2}\; j\;\omega\;\Delta\; x} - {4\;{\mathbb{e}}^{{- j}\;\omega\;\Delta\; x}} + 6 - {4\;{\mathbb{e}}^{j\;\omega\;\Delta\; x}} + {\mathbb{e}}^{2\; j\;\omega\;\Delta\; x}} = {{{2\mspace{11mu}{\cos\left( {2\;\omega\;\Delta\; x} \right)}} - {8\mspace{11mu}{\cos\left( {\omega\;\Delta\; x} \right)}} + 6} = {{2 - {4\mspace{11mu}{\sin^{2}\left( {\omega\;\Delta\; x} \right)}} - {8\mspace{11mu}{\cos\left( {\omega\;\Delta\; x} \right)}} + 6} = {{{{- 16}\;{\sin^{2}\left( \frac{\omega\;\Delta\; x}{2} \right)}\mspace{11mu}{\cos^{2}\left( \frac{\omega\;\Delta\; x}{2} \right)}} - 8 + {16\mspace{11mu}{\sin^{2}\left( \frac{\omega\;\Delta\; x}{2} \right)}} + 8} = {16\mspace{11mu}{\sin^{4}\left( \frac{\omega\;\Delta\; x}{2} \right)}}}}}} & (24)\end{matrix}$

Accordingly, amplitude characteristic can be represented as

$\begin{matrix}{{{H(\omega)}} = \frac{1}{1 + {16\mspace{11mu}\alpha\mspace{11mu}{\sin^{4}\left( \frac{\omega\;\Delta\; x}{2} \right)}}}} & (25)\end{matrix}$

On the other hand, the phase characteristic is represented asArg*H(ω)=0  (26)and it is proved that the spline filter is a phase-compensating filter.

As an example, when a filter of 50% attenuation at a cutoff frequencyω=ω_(C) is to be obtained, the amplitude characteristic may be set as

$\begin{matrix}{{H\left( \omega_{c} \right)} = \frac{1}{2}} & (27)\end{matrix}$and a constant a can be expressed as in the following equation.

$\begin{matrix}{\alpha = \frac{1}{16\mspace{11mu}{\sin^{4}\left( \frac{\omega_{c}\Delta\; x}{2} \right)}}} & (28)\end{matrix}$

The transfer characteristic (amplitude characteristic, phasecharacteristic) of the 50% attenuation filter at the cutoff frequencyω=107 _(C) will be shown in FIG. 5.

Next, solution of thus defined weighted spline filter will be describedbelow.

The matrix form of the weighted spline filter(W+αQ)S=WY  (29)has symmetric matrixM=W+αQ  (30)on the coefficient matrix at the left-side of the equation.

Decomposition of M into lower triangular matrix L and diagonal matrix Dby modified Cholesky decomposition (decomposition of the matrix M can bevery efficiently conducted since the matrix M is a sparse matrix)results inM=LDL^(T)  (31)and the weighted spline filter can be represented asLDL^(T)S=WY  (32)

WhenDL^(T)S=X  (33)thenLX=WY  (34)Since L is a lower triangular matrix, X can be easily calculated.Further, sinceL ^(T) S=D ⁻¹ X  (35)S can be easily calculated based on the calculated X.

In actually applying the present invention, the matrix M may becomesingular whenw_(k) ^(m)=0  (36)

Accordingly, it is ideally preferable that the matrix is solved bysingular value decomposition method. However, when singular valuedecomposition method is employed, a great-capacity storage and muchprocessing time are required. On the other hand, in applying the presentinvention to the actual measurement data, it is rare that the matrix Mbecomes singular, and when the matrix M is singular, it is presumed thatthe measurement data itself is defective. Therefore, Gill-Murraymodified Cholesky decomposition capable of outputting some solution evenwhen the matrix M is singular is applied to achieve both calculationefficiency and countermeasure for singular matrix.

Weighted spline filter supported by a solution can be deduced asdescribed above and a robust spline filter is obtained by repeatingcalculation until convergence condition is satisfied while updating theweight W.

FIG. 1 is a flowchart showing a first process and FIG. 2 is a functionalblock diagram of a device executing the robust spline processing, with aprocessor for spline processing 8.

In the above process, a measurement data input step for inputting ameasurement data and a selecting step (ST3) for selecting weightedspline filter formula are initially executed.

During the measurement data input step, a step ST1 for inputting themeasurement data from a measuring instrument etc. by an input unit 1 tostore the data in a storage (or memort device) 2 of a computer, and astep ST2 for deleting locally isolated singular point data contained inthe stored measurement data by a singular point data deleting unit 3 areexecuted.

In the present reference technology, the measurement data is aone-dimensional time-series data measured by a roughness-measuringmachine, where, for instance, a probe of a surface roughness tester ismoved in a direction (x-direction) to obtain a roughness data y at apredetermined pitch in x-direction. Whether a data is a singular pointdata or not can be easily determined by checking whether deviation ofthe measurement data relative to least square curve is not less than apredetermined value and not more than a predetermined width or not.

Subsequently, in the selecting step ST3, whether the measurement data isnon-periodic or periodic is judged by a judging unit 4 and weightedspline filter formula is selected in accordance with the judgment.Specifically, the formula (12) and the formula (17) are selectably usedin accordance with the judgment whether the measurement data isnon-periodic or periodic.

Next, initializing step (ST4) is conducted, where an initial value S⁰ ofthe output of the spline filtering when W=I is obtained, as illustrated(non-robust spline filtering calculation).

Next, based on the measurement data Y and S^(m) (m indicates repetitionstep), weight W^(m) is adjusted and determined by a weight adjuster 5according to below-described process (ST5).

Subsequently, spline filter output S^(m+1) is calculated based onweighted spline filter(W ^(m) +αQ)S ^(m+1) =W ^(m) Y  (37)by a spline filter output calculator 6 (ST6).

Then, below-described weight convergence judgment (ST7) is conducted bya convergence judging unit 51 and, when convergence condition is notsatisfied, m is updated (m=m+1) (ST10) to re-adjust the weight W^(m)(ST5).

When the convergence condition is satisfied (ST7: YES), the repeatingprocess is terminated to obtain the output value S^(m) (ST8) and thespline curve is outputted to an output unit 7.

In the above process, in order to adjust the weight W^(m) (ST5),adaptive biweight method is applied as follows.

$\begin{matrix}{w_{k}^{m} = \left\{ \begin{matrix}\left\lbrack {1 - \left( \frac{y_{k} - s_{k}^{m}}{\beta \cdot c} \right)^{2}} \right\rbrack^{2} & {{{y_{k} - s_{k}^{m}}} < {\beta \cdot c}} \\0 & {{{y_{k} - s_{k}^{m}}} \geq {\beta \cdot c}}\end{matrix} \right.} & (38)\end{matrix}$

Here, with reference to σ denoting standard deviation of the residualerror,

$\begin{matrix}{\beta = {{median}\left\{ {\frac{y_{k} - s_{k}^{m}}{\sigma}} \right\}}} & (39) \\{c = \left\{ \begin{matrix}6 & {\beta \leq 5} \\10 & {5 < \beta \leq 100} \\20 & {100 < \beta}\end{matrix} \right.} & (40)\end{matrix}$are set.

Also, as to the convergence condition in ST7, the repetitive process isbrought to an end at the time point that a change in the weight becamesmaller sufficiently and a formula shown below has been satisfied.

$\begin{matrix}{{\sum\limits_{k = 0}^{n - 1}{{w_{k}^{m} - w_{k}^{m - 1}}}} < {0.02 \cdot {\sum\limits_{k = 0}^{n - 1}w_{k}^{m}}}} & (41)\end{matrix}$

FIG. 3A and FIG. 3B show examples where a signal-processing method bythe robust spline filtering in the first reference technology isconducted for a one-dimensional time-series data. Here, both of a splinecurve processed by normal spline filtering and a spline curve processedby the robust spline filtering according to the present invention areapplied on the measurement data applied with a spike noise, which areshown in a superposing manner. As shown in the drawings, though theresult of normal spline filtering is influenced by the spike noise, aspline curve along original shape is obtained by the robust splinefiltering. Further, as shown in FIG. 3A, excellent trackability for ashape having gentle waviness can be obtained.

According to the present reference technology, following advantages canbe obtained.

(1) Since the spline filter can be easily robustized (i.e. made robust),the deformation at the start or end area of the measurement data can beprevented. Accordingly, the shape contained in a measurement data can beextracted without being influenced by trackability for wavinesscomponent or noise component contained in the measurement data, so thatfiltering with excellent shape trackability can be conducted and thereliability of the measurement data can be further improved.

(2) Since locally isolated singular point data contained in themeasurement data can be removed, reliability of the robust splinefiltering can be further enhanced.

(3) Since the weight becomes small in accordance with the magnitude ofdeviation of the measurement data from the spline curve calculated bythe weighted spline filter formula, robust spline filtering withoutbeing influenced by the singular point data contained in the measurementdata is possible.

(4) Since weight convergence can be judged when the variation of weightin the repetition loop process becomes smaller than a predeterminedvalue, unnecessary repetition loop process can be prevented, therebyreducing the robust spline filtering processing time.

<Second Reference Technology>

Next, second processing steps for achieving a robust spline filter willbe described below. Though the steps are the same as the firstprocessing steps, different calculation formulae are used therein.

Specifically, the weighted spline filter formula(W+αQ)S=WY  (42)is modified as(I+αQ)S=WY+(I−W)S  (43)Here,(I+αQ)S ^(m+1) =W ^(m) Y+(I−W ^(m))S ^(m)  (44)is used in repetition step m.

In the second processing step, following advantage as well as theadvantages (1) to (4) in the first processing step can be expected.

(5) Since the left-side coefficient matrixI+αQ  (45)constantly represents the same value during the repetition step, totalrobust spline filtering processing time can be reduced.<Third Reference Technology>

Next, a signal-processing method for a measurement data as atwo-dimensional data measured two-dimensionally will be described belowas a third reference technology of the signal-processing method of thepresent invention. Here, the measurement data as a two-dimensional datarefers to, for instance, (x, y) coordinates value etc. obtained bymeasuring contour curve of a workpiece under constant z-coordinate by apredetermined pitch using a coordinate measuring machine and the like ora data obtained by importing a figure drawn on a plane with a scanner.In other words, though the object to be processed is only y-coordinatein the first reference technology, both of x-coordinate and y-coordinateare processed in the third reference technology.

The basic arrangement of the third reference technology is the same asthe first reference technology, which is characterized in a formulacorresponding to the formula (6) in starting the calculation of splinecurve s.

In the third reference technology, under the condition that the splineenergy is minimized, a spline curve for minimizing sum of squares of theX and Y-direction distance between the measurement data (x_(k), y_(k))and point (s_(x)(x_(k), y_(k)), s_(y)(x_(k), y_(k))) on the spline curves corresponding to the measurement data (x_(k), y_(k)) is calculated. Inother words, under the above incidental condition, a spline curve scapable of minimizing I(s) represented by the following formula iscalculated.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}\;{w_{k}\left\lbrack {\left\{ {x_{k} - {s_{x}\left( {x_{k},y_{k}} \right)}} \right\}^{2} + \left\{ {y_{k} - {s_{y}\left( {x_{k},y_{k}} \right)}} \right\}^{2}} \right\rbrack}} + \mspace{65mu}{\alpha{\sum\limits_{k = 0}^{n - 1}\;\left\{ {{\nabla_{x}^{2}{s\left( {x_{k},y_{k}} \right)}} + {\nabla_{y}^{2}{s\left( {x_{k},y_{k}} \right)}}} \right\}}}}} & (46)\end{matrix}$

In the above, secondary approximation of laplacian on the second term ofright side is represented as follows:∇_(x) ² s(x _(k) ,y _(k))=s _(x)(x _(k+1))−2s _(x)(x _(k))+s _(x)(x_(k−1))∇_(y) ² s(x _(k) ,y _(k))=s _(y)(y _(k+1))−2s _(y)(y _(k))+s _(y)(y_(k−1))  (47)

Then, weighted spline filter described in the first reference technologyis applied for each x component and y component (see formula (37)).

In the above, the constant α can be calculated based on a sampling pitchΔ1 along measurement path and a cutoff wavelength λc′ according to thefollowing formula:

$\begin{matrix}{\alpha = \frac{1}{16\mspace{14mu}{\sin^{4}\left( \frac{{\pi \cdot \Delta}\; l}{\lambda^{\prime}c} \right)}}} & (48)\end{matrix}$

Then, a spline filter for deducing the spline curve for each section ofthe two-dimensional measurement data can be constructed.

Further, in a robust spline filter where the processing is repeated byupdating the weight W until the convergence condition (formula (41)) issatisfied, (y_(k)−s_(k) ^(m)) in formula (38) is set as a distancebetween two points as represented in the following formula, whichspecifically is a distance between the measurement data (x_(k), y_(k))and the point (s_(x)(x_(k), y_(k)), s_(y)(x_(k), y_(k))) on the splinecurve s corresponding to the measurement data (x_(k), y_(k)).d _(k) =√{square root over({x_(k)−s_(x)(x_(y),y_(k))}²+{y_(k)−s_(y)(x_(k),y_(k))}²)}{square rootover ({x_(k)−s_(x)(x_(y),y_(k))}²+{y_(k)−s_(y)(x_(k),y_(k))}²)}  (49)

The convergence of the weight W calculated by the formula (38) applyingthe formula (49) is judged based on the formula (41). When the weight Wis converged, a spline curve corresponding to measurement data iscalculated based on the output value S^(m) (spline function). The splinecurve is outputted to the output unit.

FIG. 4A shows a comparison of a result of spline processing on an inputdata applying spike nose on a folium and a result of robust splineprocessing thereon. As shown in FIG. 4A, though the result by theordinary spline processing is influenced by the spike noise, robustresult restraining the spike noise can be obtained by the robust splineprocessing. FIG. 4B shows a result of spline processing on an input dataapplying spike noise on an airfoil and a result of robust splineprocessing thereon, which exhibits the same result as FIG. 4A.

According to the third reference technology, following advantages aswell as the advantages (1) to (5) in the first and the second referencetechnology can be obtained.

(6) When the measurement data is two-dimensional data on an orthogonalcoordinates, since the deviation of the measurement data from the splinecurve is determined based on the sum of squares of the components forthe respective axes (for instance, X-axis component and Y-axiscomponent), the deviation can be easily calculated. Accordingly, theweight of the respective measurement data can be easily determined.

(7) Even when the measurement data is two-dimensional data, since thespline filter output can be obtained based on the result of weightedspline filtering for the respective axes components (for instance,X-axis component and Y-axis component), the calculation process can besimplified even for a complicated curve, so that the time required forrobust spline filtering calculation on a measurement data can bereduced.

(8) When two-dimensional data is obtained by profiling a workpiece on atwo-dimensional plane to input as a measurement data, since themeasurement data is inputted at a predetermined interval along themeasurement path, shape-changing point (e.g. changing point fromstraight line to arc or boundary point of a step) can be more accuratelyrecognized as compared to an arrangement where the measurement data isinputted at a predetermined interval in X-axis direction. In otherwords, shape judgment error can be prevented and highly reliablemeasurement data can be inputted.

<Fourth Reference Technology>

Next, a signal-processing method for a measurement data as athree-dimensional data measured three-dimensionally will be describedbelow as a fourth reference technology of the signal-processing methodof the present invention. Here, the measurement data as athree-dimensional data refers to, for instance, (x, y, z) coordinatesvalue etc. obtained by measuring a surface of a workpiece at apredetermined pitch using a coordinate measuring machine and the like.In other words, though the object to be processed is only y-coordinatein the first reference technology, all of x-coordinate, y-coordinate andz-coordinate are processed in the fourth reference technology.

The basic arrangement of the fourth reference technology is the same asthe first reference technology, which is characterized in a formulacorresponding to the formula (6) in starting the calculation of splinecurve s.

In the fourth reference technology, under the condition that the splineenergy is minimized, a spline curve for minimizing sum of squares of theX, Y and Z-direction distance between the measurement data (x_(k),y_(k), z_(k)) and point (s_(x)(x_(k), y_(k), z_(k)), s_(y)(x_(k), y_(k),z_(k)), s_(z)(x_(k), y_(k), z_(k))) on the spline curve s correspondingto the measurement data (x_(k), y_(k), z_(k)) is calculated. In otherwords, under the above incidental condition, a spline curve s capable ofminimizing I(s) represented by the following formula is calculated.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}\;\left\lbrack {\left\{ {x_{k} - {s_{x}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2} + \left\{ {y_{k} - {s_{y}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2} + \left\{ {z_{k} - {s_{k}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2}} \right\rbrack} + {\alpha{\sum\limits_{k = 0}^{n - 1}\;\left\{ {{\nabla_{x}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}} + {\nabla_{y}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}} + {\nabla_{z}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}}} \right\}}}}} & (50)\end{matrix}$

In the above, secondary approximation of laplacian on the second term ofright side is represented as in the third reference technology.

Then, weighted spline filter described in the first reference technologyis applied for each x, y and z component (see formula (37)).Incidentally, the constant α is defined based on a sampling pitch Δ1along measurement path in the three-dimensional space and cutoffwavelength λ_(c)′ according to the formula (48).

Then, a spline filter for deducing the spline curve for each section ofthe three-dimensional measurement data can be constructed.

Further, in a robust spline filter where the processing is repeated byupdating the weight W until the convergence condition (formula (41)) issatisfied, (y_(k)−s_(k) ^(m)) in formula (38) is set as a distancebetween two points as represented in the following formula, whichspecifically is a distance between the measurement data (x_(k), y_(k),z_(k)) and point (s_(x)(x_(k), y_(k), z_(k)), s_(y)(x_(k), y_(k),z_(k)), s_(z)(x_(k), y_(k), z_(k))) on the spline curve s correspondingto the measurement data (x_(k), y_(k), z_(k)).d _(k) =√{square root over({x_(k)−s_(x)(x_(k),y_(k),z_(k))}²+{y_(k)−s_(y)(x_(k),y_(k),z_(k))}²+{z_(k)−s_(z)(x_(k),y_(k),z_(k))}²)}{squareroot over({x_(k)−s_(x)(x_(k),y_(k),z_(k))}²+{y_(k)−s_(y)(x_(k),y_(k),z_(k))}²+{z_(k)−s_(z)(x_(k),y_(k),z_(k))}²)}{squareroot over({x_(k)−s_(x)(x_(k),y_(k),z_(k))}²+{y_(k)−s_(y)(x_(k),y_(k),z_(k))}²+{z_(k)−s_(z)(x_(k),y_(k),z_(k))}²)}  (51)

The convergence of the weight W calculated by the formula (38) applyingthe formula (51) is judged based on the formula (41). When the weight Wis converged, a spline curve corresponding to measurement data iscalculated based on the output value S^(m) (spline function). The splinecurve is outputted to the output unit.

According to the fourth reference technology, following advantage aswell as the advantages (1) to (5) of the advantages of the first and thesecond reference technologies can be obtained.

(9) The advantages (6) to (8) of the third reference technology can befurther achieved in a three-dimensional data. Accordingly, when themeasurement data is a three-dimensional data, calculation load can bereduced without increasing processing time of the robust splinefiltering.

<Reference Variant>

A modification of the present invention will be described below. Thoughthe spline curve at the time of convergence judgment in the firstreference technology is directly outputted as a result ofsignal-processing, the spline curve is re-calculated and the result isoutputted as the signal-processing result in the present modification.

FIG. 6 shows a modification of spline curve output (ST9) of FIG. 1.

Initially, the calculated output value S^(m) is inputted (ST91).Subsequently, whether re-calculation is to be conducted or not isdetermined (ST92). For instance, “YES” is selected when highly accuratesignal-processing result is to be obtained and “NO” is selected when itis considered that sufficient accuracy has been obtained by an operatorat the time, or in advance.

When the re-calculation is not conducted (NO), the spline curve of theoutput value S^(m) is outputted by the output unit 7. When there-calculation is to be conducted (YES), weight exceeding apredetermined value designated in advance is updated to 1 and weight notgreater than a predetermined value is updated to 0 (ST93). In otherwords, the measurement data having weight exceeding the predeterminedvalue is judged effective and 100% contribution on the spline processingis set thereon and the measurement data having weight not greater thanthe predetermined value is judged inefficient and 0% contribution isset.

Subsequently, weighted spline filtering is conducted based on theupdated weight to obtain an output (ST94). The obtained spline curve isoutputted by the output unit 7 as a signal-processing result (ST95).

The reference variant can be applied on all of the first to fourthreference technologies, where the following advantage as well as theadvantages (1) to (9) can be obtained.

(10) The weight is updated to 1 when the weight at the time when theweight is judged converged exceeds a predetermined value, and the splinefilter output is obtained again to output the result as thesignal-processing result. In other words, when the weight is judgedconverged after repeating the weight adjusting step and the splinefilter output calculating step, the measurement data at the point wherethe weight exceeds the predetermined value is considered effective toupdate the weight thereof to 1, and the spline filter output can bere-calculated, so that the robust spline processing for the measurementdata can be more securely conducted. Further, since the result isoutputted as the signal processing result, a spline curve havingsufficiently small error relative to the original shape componentcontained in the measurement data can be obtained, thus conductingrobust spline filter processing with excellent shape trackability.

<First Embodiment>

Next, the signal processing method for the curved surface data measuredin the surface area will be explained as a first relating to the signalprocessing method of the present invention.

Herein, the so-called curved surface data signifies, for example,surface roughness data or form data etc. obtained by scanning thesurface of the object to be measured by the coordinate measuring machineor the surface texture measuring machine.

And, this first embodiment relates to a method of filter-processingmeasurement data f_(i) (x_(i), y_(i)) of the surface at a samplingposition (x_(i), y_(j)), thereby to obtain a curved surface z=s (x, y)having the disturbance component or the high-frequency componentremoved.

A basic processing procedure of the first embodiment is similar to thatof the first reference technology or the second reference technology;however its special feature lies in an formula of a bending energy ofthe spline and so on, which is caused by the fact that data to beapplied is curved surface data.

A sum of squares of the weighted residual between the measurement dataf_(i) (x_(i), y_(i)) of the surface at the sampling position (x_(i),y_(i)) and the curved surface z=s (x, y) to be obtained after the filterprocess is expressed by the following formula with w_(i) assumed to be aweight for the residual.

$\begin{matrix}{\sum\limits_{i = 0}^{n - 1}\;{w_{i} \cdot \left\{ {f_{i} - {s_{i}\left( {x_{i},y_{i}} \right)}} \right\}^{2}}} & (52)\end{matrix}$

Also, the bending energy in a case where the surface z=s (x, y) has beenthought to be a thin plate is expressed by the following formula.

$\begin{matrix}{\int{\int{{\left\{ {\left( \frac{\partial^{2}s}{\partial x^{2}} \right)^{2} + \left( \frac{\partial^{2}s}{\partial y^{2}} \right)^{2} + {2\left( \frac{\partial^{2}s}{{\partial x}{\partial y}} \right)^{2}}} \right\} \cdot {\mathbb{d}x}}{\mathbb{d}y}}}} & (53)\end{matrix}$

Under the constraint condition that this bending energy formula (53) isminimized, obtaining s (x, y) for minimizing the sum-of-squares formula(52) of the weighted residual causes the weighted spline filter (robustspline filter) for the surface area to be defined. That is, it isexpressed by the following formula with λ assumed to be a Lagrangianundetermined multiplier.

$\begin{matrix}\left. {{\sum\limits_{i = 0}^{n - 1}\;{w_{i} \cdot \left\{ {f_{i} - {s\left( {x_{i},y_{i}} \right)}} \right\}^{2}}} + {\lambda{\int{\int{{\left\{ {\left( \frac{\partial^{2}s}{\partial x^{2}} \right)^{2} + \left( \frac{\partial^{2}s}{\partial y^{2}} \right)^{2} + {2\left( \frac{\partial^{2}s}{{\partial x}{\partial y}} \right)^{2}}} \right\} \cdot {\mathbb{d}x}}{\mathbb{d}y}}}}}}\rightarrow\min\limits_{s{({x_{i},y_{k}})}} \right. & (54)\end{matrix}$

As mentioned above, the weighted spline filter on the measurement dataof the surface area has been defined. Thereupon, by solving the weightedspline filter repeatedly while updating the weight for the residualaccording to the Biweight method it is made possible to obtain therobust spline filter on the measurement data of the surface area.Additionally, the computation for updating the weight is carried outwith the formula (38) to the formula (41) explained in the firstreference technology. And, such processes are realized with theflowchart of FIG. 13 and the functional block diagram of FIG. 2.

As one example, a case will be explained of signal-processing themeasurement data of the surface, which has been sampled in a latticeshape. The sampling intervals in the x direction and in the y directionare assumed to be Δx and Δy respectively, and the data numbers in the xdirection and in the y direction are assumed to be nx and nyrespectively. And, upon assuming height data at a sampling position(x_(k), y_(l)) to be f_(k,I), an formula I (s) for defining the weightedspline filter is expressed by the following expression.

$\begin{matrix}{{I(s)} = {{\sum\limits_{l = 0}^{{ny} - 1}\;{\sum\limits_{k = 0}^{{nx} - 1}\;{w_{k,l} \cdot \left( {f_{k,l} - s_{k,l}} \right)^{2}}}} + \mspace{76mu}{\lambda{\int{\int{{\left\{ {\left( \frac{\partial^{2}s}{\partial x^{2}} \right)^{2} + \left( \frac{\partial^{2}s}{\partial y^{2}} \right)^{2} + {2\left( \frac{\partial^{2}s}{{\partial x}{\partial y}} \right)^{2}}} \right\} \cdot {\mathbb{d}x}}{\mathbb{d}y}}}}}}} & (55)\end{matrix}$

Herein, digitizing the second term yields the following formula.

$\begin{matrix}{{{I(s)} = {{\sum\limits_{l = 0}^{{ny} - 1}\;{\sum\limits_{k = 0}^{{nx} - 1}\;{w_{k,l} \cdot \left( {f_{k,l} - {sx}_{k,l}} \right)^{2}}}} + \mspace{70mu}{{\lambda\Delta}\; x\;\Delta\; y{\sum\limits_{l = 0}^{{ny} - 1}\;{\sum\limits_{k = 0}^{{nx} - 1}\;\left\{ {\left( {\nabla_{x}^{2}s} \right)^{2} + \left( {\nabla_{y}^{2}s} \right)^{2} + {2\left( {\nabla_{xy}^{2}s} \right)^{2}}} \right\}}}}}}{{\nabla_{x}^{2}{s\left( {x_{k},y_{l}} \right)}} = \frac{s_{{k + 1},l} - {2s_{k,l}} + s_{{k - 1},l}}{\Delta\; x^{2}}}{{\nabla_{y}^{2}{s\left( {x_{k},y_{l}} \right)}} = \frac{s_{k,{l + 1}} - {2s_{k,{l + 1}}} + s_{k,{l - 1}}}{\Delta\; y^{2}}}{{\nabla_{xy}^{2}{s\left( {x_{k},y_{l}} \right)}} = \frac{s_{{k + 1},{l + 1}} - s_{k,{l + 1}} - s_{{k + 1},l} + s_{k,l}}{\Delta\; x\;\Delta\; y}}} & (56)\end{matrix}$

Herein, boundary conditions are given by the following formulas (57).∇_(x) ² s(x ₀ ,y _(l))=∇_(x) ² s(x _(n) _(x) ⁻¹ ,y _(l))=0 l=0,1, . . ., n _(y)−1∇_(y) ² s(x _(k) ,y ₀)=∇_(y) ² s(x _(k),y_(n) _(y) ⁻¹)=0 k=0,1, . . . ,n _(x)−1∇_(xy) ² s(x ₀ ,y _(l))=∇_(xy) ² s(x _(n) _(x) ⁻¹ ,y _(l))=0 l=0,1, . .. , n _(y)−1∇_(xy) ² s(x _(k) ,y ₀)=∇_(xy) ² s(x _(k),y_(n) _(y) ⁻¹)=0 k=0,1, . . ., n _(x)−1  (57)

Then, the following matrix expression is obtained.(W+Q)S=WF  (58)

Herein, W is an n×n diagonal matrix with a weight w_(i) assumed to be adiagonal element, Q is an n×n coefficient matrix, S is an n-dimensionalfilter output vector, and F is an n-dimensional filter input vector,where n=nx×ny.

In the above-mentioned formula (58), making a solution thereof for Swith W set to W=I (unit matrix) allows the output of the spline filterrelating to the surface area to be obtained. This is equivalent to ST 14in FIG. 13.

Also, by making a solution repeatedly while updating the weight matrixit is made possible to obtain the output of the robust spline filter forthe surface area. This is equivalent to ST 15, ST 16, ST 17, and ST 20in FIG. 13.

In FIG. 7 to FIG. 11, results obtained by applying the robust splinefilter on the measurement data of the surface area are shown.

FIG. 7 is a view illustrating measurement data obtained in themeasurement, and FIG. 8 is a view illustrating a result obtained byapplying the robust spline filter on the measurement data of FIG. 7.And, in FIG. 9, a result obtained by applying the Gaussian filter on themeasurement data of FIG. 7 is shown as a comparative example.

Also, in FIG. 10, a sectional profile along the X direction of FIG. 7 isshown, and in FIG. 11, a sectional profile along the Y direction of FIG.7 is shown.

As shown in the profiles of FIG. 10 or FIG. 11, it is seen that thesignal process by the robust spline filter allows an excellent result ofhaving exhibited trackability for the measurement data to be obtained.

Herein, the specific solving method for the curved surface data, whichhas been sampled in a lattice shape, has been shown; however the curvedsurface data such that a triangular mesh structure is formed can be alsosolved by employing a finite element method etc.

Next, a filter characteristic of the robust spline filter (splinefilter) on the measurement data of the surface area will be explainedbriefly.

Expressing a transfer function in a stationary state in terms of a ztransformation with the weight W set to W=1 yields the following.

$\begin{matrix}{{{H\left( {z_{x},z_{y}} \right)} = {\frac{s_{k,l}}{z_{k,l}} = \frac{1}{1 + {\lambda_{x}H_{x}} + {\lambda_{y}H_{y}} + {\lambda_{xy}H_{xy}}}}}{where}{{H_{x}\left( z_{x} \right)} = {z_{x}^{2} - {4z_{x}} + 6 - {4z_{x}^{- 1}} + z_{x}^{- 2}}}{{H_{y}\left( z_{y} \right)} = {z_{y}^{2} - {4z_{y}} + 6 - {4z_{y}^{- 1}} + z_{y}^{- 2}}}{{H_{xy}\left( {z_{x},z_{y}} \right)} = {{z_{x}z_{y}} + {z_{x}z_{y}^{- 1}} + {z_{x}^{- 1}z_{y}} + {z_{x}^{- 1}z_{y}^{- 1}} - {2z_{x}} - {2z_{y}} + 4 - {2z_{x}^{- 1}} - {2z_{y}^{- 1}}}}} & (59)\end{matrix}$

Herein, upon assuming frequencies in the x direction and in the ydirection to be ω_(x), and ω_(y) respectively, z_(x) and z_(y) areexpressed as follows respectively.z _(x) =e ^(jω) ^(x) ^(·Δx)z _(y) =e ^(jω) ^(y) ^(·Δy)  (60)

Accordingly, the formula (59) is expressed as follows.

$\begin{matrix}{{{H\left( {\omega_{x},\omega_{y}} \right)} = \frac{1}{1 + {\lambda_{x}16{\sin^{4}\left( \frac{{\omega_{x} \cdot \Delta}\; x}{2} \right)}} + {\lambda_{y}16{\sin^{4}\left( \frac{{\omega_{y} \cdot \Delta}\; y}{2} \right)}} + {\lambda_{xy}16{\sin^{2}\left( \frac{{\omega_{x} \cdot \Delta}\; x}{2} \right)}{\sin^{2}\left( \frac{{\omega_{y} \cdot \Delta}\; y}{2} \right)}}}}{additionally}{{H_{x}\left( \omega_{x} \right)} = {16{\sin^{4}\left( \frac{{\omega_{x} \cdot \Delta}\; x}{2} \right)}}}{{H_{y}\left( \omega_{y} \right)} = {16{\sin^{4}\left( \frac{{\omega_{y} \cdot \Delta}\; y}{2} \right)}}}{{H_{xy}\left( {\omega_{x},\omega_{y}} \right)} = {16{\sin^{2}\left( \frac{{\omega_{x} \cdot \Delta}\; x}{2} \right)}{\sin^{2}\left( \frac{{\omega_{y} \cdot \Delta}\; y}{2} \right)}}}} & (61)\end{matrix}$

Herein, in the cut-off frequency ω_(c), so as to obtain a 50%attenuation characteristic, the transfer function is put as follows.

$\begin{matrix}{{H\left( {\omega_{c},0} \right)} = {{H\left( {0,\omega_{c}} \right)} = {{H\left( {\frac{\omega_{c}}{\sqrt{2}},\frac{\omega_{c}}{\sqrt{2}}} \right)} = \frac{1}{2}}}} & (62)\end{matrix}$

Then, it follows that the followings are obtained, whereby it can beunderstood that this filter is a phase compensating filter without adelay.

$\begin{matrix}\begin{matrix}{\lambda_{x} = \frac{1}{16{\sin^{4}\left( \frac{{\omega_{c} \cdot \Delta}\; x}{2} \right)}}} \\{\lambda_{y} = \frac{1}{16{\sin^{4}\left( \frac{{\omega_{c} \cdot \Delta}\; y}{2} \right)}}} \\{\lambda_{xy} = \frac{1 - {16\lambda_{x}{\sin^{4}\left( \frac{{\omega_{c} \cdot \Delta}\; x}{2\sqrt{2}} \right)}} - {16\lambda_{y}{\sin^{4}\left( \frac{{\omega_{c} \cdot \Delta}\; y}{2\sqrt{2}} \right)}}}{16{\sin^{2}\left( \frac{{\omega_{c} \cdot \Delta}\; x}{2\sqrt{2}} \right)}{\sin^{2}\left( \frac{{\omega_{c} \cdot \Delta}\; y}{2\sqrt{2}} \right)}}}\end{matrix} & (63)\end{matrix}$

Herein, it is assumed that the sampling intervals in the x direction andin the y direction are equal, a product of the frequency ω and thesampling interval is sufficiently small within an all-frequency rangewhere the input data exists, and the following approximation formulaholds.sin(ω·Δt)≈ω·ΔthereinΔx=Δy=Δt  (64)

Then, the following formula holds, whereby it can be seen that assuminga sufficiently small sampling interval allows the above filter to becomea filter having an isotropic transfer function practically.

$\begin{matrix}{{H\left( {\omega_{x},\omega_{y}} \right)} \approx \frac{\omega_{c}^{4}}{\omega_{c}^{4} + \left( {\omega_{x}^{2} + \omega_{y}^{2}} \right)^{2}}} & (65)\end{matrix}$

In FIG. 12, the transfer characteristic at the time that Δx=Δy=Δz hasbeen put is shown.

As mentioned above, the preferred embodiment of the present inventionhas been provided for explanation; however the present invention is notrestricted to this embodiment described above, and various modificationsmay be done without departing from the spirit of the present invention.

The present invention, which is not restricted to each of theseembodiments, can be carried out for, for example, all ofthree-dimensional roughness data, measurement data by the contourmeasuring machine, data measured in the roundness measuring machine,form data measured in the coordinate measuring machine, data measured inthe image measuring machine, etc.

Also, irrespective of types of sensors to be employed, i.e. a contacttype sensor or non-contact type sensor to be employed for collecting themeasurement data, and further, without being limited to the measurementdata being a surface texture data of the object to be measured, thepresent invention can be carried out for electric signal data etc. aswell, which occurs in a time-series manner.

Further, in this embodiment, only the case where the measurement datahas been once filed in the memory device has been explained; however thepresent invention can be carried out also in a case of performing thecomputing process whenever the measurement data is collected, namely, inwhat is called a real time basis.

Further, a signal processing program for a computer, which executes thesignal processing method of the present invention, may be employed, andthis signal processing program can be filed in a form, which isexecutable by various computers, by employing a portable-type storagemedium such as a CD-ROM. Also, this signal processing program may be ofcompile format to be translated into a machine language, or may be ofinterpreter format to be translated into an intermediate language.

Also, causing the computer to execute the signal processing programenables the signal processing unit to be configured. That is, by causingthe computer to execute the measurement data inputting step, theselecting step, the initializing step, the weight adjusting step, thespline filter output calculating step, the convergence determining step,and the outputting step respectively, the measurement data inputtingdevice, the selector, the initializing device, the weight adjustingdevice, the spline filter output calculator, the converge judging unit,and the outputting device are configured respectively, thereby allowingthe signal processing unit to be configured.

The priority application Number JP2004-010918 upon which this patentapplication is based is hereby incorporated by reference.

1. A signal processing method for filtering a measurement data obtainedby measuring a surface area having a predetermined size, comprising thesteps of: inputting the measurement data; selecting a predeterminedweighted-spline filter formula; applying a weight on the measurementdata by a unit matrix so as to obtain an initial value of a splinefilter output; adjusting and determining the weight on the measurementdata; calculating the spline filter output using the determined weight;determining convergence of the weight; and outputting a signalprocessing result based on the spline filter output, wherein, when it isnot judged that the weight is converging, updating the weight andrepeating the steps of adjusting the weight and calculating the splinefilter output, and wherein a smoothing spline curved surfacecorresponding to the measurement data is generated so as to conduct arobust spline filtering on the measurement data.
 2. The signalprocessing method according to claim 1, wherein the determined weight isadjusted to a smaller value as a deviation of the measurement data fromthe generated spline curve by the weighted-spline filter formulaincreases.
 3. The signal processing method according to claim 1,wherein, the weight is converging when an amount of weight-change in theweight adjusting step becomes a predetermined value or less.
 4. Thesignal processing method according to claim 1, wherein the output stepcomprises the further steps of: updating the weight to 1 when the weightof the measurement data exceeds a predetermined value; re-calculatingthe spline filter output based upon the updated weight; and outputtingthe re-calculated spline filter output as a signal processing result. 5.The signal processing method according to claim 1, wherein the step ofinputting measurement data includes deleting a locally isolated singularpoint data relative to the measurement data.
 6. A signal processingprogram executing a signal processing method for filtering a measurementdata obtained by measuring a surface area having a predetermined size,with a computer, the method comprising the steps of: inputting themeasurement data; selecting a predetermined weighted-spline filterformula; applying a weight on the measurement data by a unit matrix soas to obtain an initial value of a spline filter output; adjusting anddetermining the weight on the measurement data; calculating the splinefilter output using the determined weight; determining convergence ofthe weight; and outputting a signal processing result based on thespline filter output, wherein, when it is not judged that the weight isconverging, updating the weight and repeating the steps of adjusting theweight and calculating the spline filter output, and wherein a smoothingspline curved surface corresponding to the measurement data is generatedso as to conduct a robust spline filtering on the measurement data.
 7. Arecording medium storing a signal processing program therein, whereinthe program executes a signal processing method for filtering ameasurement data obtained by measuring a surface area having apredetermined size, with a computer, the method comprising the steps of:inputting the measurement data; selecting a predeterminedweighted-spline filter formula; applying a weight on the measurementdata by a unit matrix so as to obtain an initial value of a splinefilter output; adjusting and determining the weight on the measurementdata; calculating the spline filter output using the determined weight;determining convergence of the weight; and outputting a signalprocessing result based on the spline filter output, wherein, when it isnot judged that the weight is converging, updating the weight andrepeating the steps of adjusting the weight and calculating the splinefilter output, and wherein a smoothing spline curved surfacecorresponding to the measurement data is generated so as to conduct arobust spline filtering on the measurement data.
 8. A signal processingunit executing a signal processing program with a computer, wherein theprogram executes a signal processing method for filtering a measurementdata obtained by measuring a surface area having a predetermined size,with a computer, the method comprising the steps of: inputting themeasurement data; selecting a predetermined weighted-spline filterformula; applying a weight on the measurement data by a unit matrix soas to obtain an initial value of a spline filter output; adjusting anddetermining the weight on the measurement data; calculating the splinefilter output using the determined weight; determining convergence ofthe weight; and outputting a signal processing result based on thespline filter output, wherein, when it is not judged that the weight isconverging, updating the weight and repeating the steps of adjusting theweight and calculating the spline filter output, and wherein a smoothingspline curved surface corresponding to the measurement data is generatedso as to conduct a robust spline filtering on the measurement data.